We show that if G is a K r -free graph on N, there is an independent set in G which contains an arbitrarily long arithmetic progression together with its difference. This is a common generalization of theorems of Schur, van der Waerden, and Ramsey. We also discuss various related questions regarding
Weakly arithmetic progressions in sets of natural numbers
β Scribed by Egbert Harzheim
- Publisher
- Elsevier Science
- Year
- 1991
- Tongue
- English
- Weight
- 186 KB
- Volume
- 89
- Category
- Article
- ISSN
- 0012-365X
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## Abstract The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type Ο, where an arithmetic progression of order type Ο means an increasing sequence of ordinal numbers (Γ + δγ)Ξ³<Ξ³<>r, Ξ΄ β 0.
proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, ..., N} is at least cN 1/4 , and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N] d
A subset of the natural numbers is k-sum-free if it contains no solutions of the equation x 1 + } } } +x k = y, and strongly k-sum-free when it is l-sum-free for every l=2, ..., k. It is shown that every k-sum-free set with upper density larger than 1Γ(k+1) is a subset of a periodic k-sum-free set a
We construct a universal r.e. set in the following manner: For any (n, x) we construct a set Un,, E 8 such that the set of all (z, n, x ) such that z E U,,,, is r.e. We construct the set Un,x by steps, and on step s we build a finite approximation U,,.x,s of U,,,,, and finally we take Let us describ